Real Numbers — NCERT Class 10 Complete Lesson
?? Euclid's Division Lemma
For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
Example: For a = 455 and b = 42: 455 = 42 × 10 + 35 ? (remainder 35 < 42)
This lemma is used to find the HCF (Highest Common Factor) of two numbers.
?? The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes, and this factorisation is unique (except for the order of factors).
No matter how you factorise 12, you always get the same prime factors!
Use this to find HCF and LCM:
HCF = product of smallest powers of common primes
LCM = product of greatest powers of all primes
v2 is Irrational — Proof
Assume v2 = p/q (in lowest terms). Then 2 = p²/q², so p² = 2q². This means p² is even, so p is even. Let p = 2m. Then 4m² = 2q², so q² = 2m², meaning q is also even. But then p and q have a common factor 2 — contradiction! Hence v2 is irrational.
e.g., 3 + v2 is irrational. 5v3 is irrational.
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